@suresh: Somehow to me, the worst case for anti-concentration about the mean i.e. $0$ seemed to be when all the $a_i$'s are the same i.e. about $n/w$. In case $w\gg n$, then all the $a_i$'s can be like $n/w$ and then the anti-concentration will only be about $n/\sqrt{w}$ which will go to $0$ as $w$ becomes larger. Anyway, in my case $w \ll n$.
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Anindya DeMar 7 '11 at 17:06